If I have some solid material like biomass and incinerate it at 1000 Celsius degrees for 15 minutes in an oxidized atmosphere within an incineration oven. As an output it gives me ash. Is the incineration complete or incomplete? Like for example we have incomplete combustion (lack of oxygen) and complete combustion (enriched oxygen medium). What about this incineration?
The comment by Maurice is perfectly good for analytical purposes.
For commercial use, in practical applications, the ash residue is not the only product of combustion that must be completely oxidized. The gaseous products should be $CO_2$, $H_2O$ and $N_2$, maybe a little $SO_2$ - but sometimes organic compounds volatilize before burning and escape. In backyard leaf-burning, or in fireplace fires, this escape is mostly innocuous (except for creosote buildup in chimneys), but when tons of garbage are pyrolyzed, the gaseous products can be obnoxious. In some cases, extra fuel is added to a pyrolytic garbage treatment to raise the temperature of burning and reduce noxious emissions, but that increases the cost.
So if you are going to ask about complete combustion, you might analyze for $CO$, nitrogen oxides, and other volatiles that may have escaped.